Abstract: We generalise various non-triviality conditions for group actions to Fell bundles over discrete groups and prove several implications between them. We also study sufficient criteria for the reduced section C*-algebra C_r(B) of a Fell bundle (B_g) to be strongly purely infinite. If the unit fibre A=B_e contains an essential ideal that is separable or of Type I, then the Fell bundle is aperiodic if and only if it is topologically free. If, in addition, G=Z or G=Z/p for a square-free number p, then these equivalent conditions are satisfied if and only if A detects ideals in C_r(B), if and only if A^+ \ {0} supports C_r(B) in the Cuntz sense. For G as above and arbitrary A, C_r(B) is simple if and only if the Fell bundle B is minimal and pointwise outer. In general, B is aperiodic if and only if each of its non-trivial fibres has a non-trivial Connes spectrum. If G is finite or if A contains an essential ideal that is of Type I or simple, then aperiodicity is equivalent to pointwise pure outerness.

Abstract: We investigate the relation between the dynamics of a single oscillator with delayed self-feedback and a feed-forward ring of such oscillators, where each unit is coupled to its next neighbor in the same way as in the self-feedback case. We show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring. In particular, if for the single oscillator the periodic solution is resonant to the delay, it can be embedded into a ring with instantaneous couplings. We discover several cases where the stability of a periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring. As a specific example we demonstrate how the complex bifurcation scenario of simultaneously emerging multi-jittering solutions can be transferred from a single oscillator with delayed pulse feedback to multi-jittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling. Finally, we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of FitzHugh-Nagumo type.

Abstract: We construct an example of a unit-regular ring which is not strongly clean, answering an open question of Nicholson. We also characterize clean matrices with a zero column, and this allows us to describe an interesting connection between unit-regular elements and clean elements. It is also proven that given an element $a$ in a ring $R$, if $a,a^2,\ldots, a^k$ are all regular elements in $R$ (for some $k\geq 1$), then there exists $w\in R$ such that $a^{i}w^{i}a^{i}=a^{i}$ for $1\leq i\leq k$, and a similar statement holds for unit-regular elements. The paper ends with a large number of examples elucidating further connections (and disconnections) between cleanliness, regularity, and unit-regularity.

Abstract: Let Γ be a torsionless grading monoid, R=⊕α∈ΓRα a Γ-graded integral domain, H the set of nonzero homogeneous elements of R, K the quotient field of R0, and G0 = Γ∩−Γ the group of units of Γ. We say that R is a graded-valuation domain if either x∈R or x−1∈R for every nonzero homogeneous element x∈RH. In this paper, we show that R is a graded-valuation domain if and only if Γ is a valuation monoid,Rα = Kx for every 0≠x∈Rα whenever α is not a unit of Γ, andT=⊕α∈G0Rα is a graded-valuation domain.Let R = Kγ[X;Γ] be a twisted semigroup ring of Γ over K, C a totally ordered (additive) abelian group, C′ a subgroup of C, μ:K→C′∪{∞} a valuation, φ:Γ→C a function such that C′∪φ(Γ) generates C, and v:R→C∪{∞} the function defined by v(∑aαXα)=inf{μ(aα)+φ(α)} for every ∑aαXα∈R. We show that v is a valuation if and only if μ(γ(a,b))+φ(a+b) = φ(a)+φ(b) for every a,b∈Γ.

Abstract: For each let be the affine group over the integers. For every point let Let be the subgroup of the additive group generated by . If then . Thus, is a complete classifier of . By contrast, if , knowledge of alone is not sufficient in general to uniquely recover ; as a matter of fact, determines precisely different orbits, where is the denominator of the smallest positive non-zero rational in and is the Euler function. To get a complete classification, rational polyhedral geometry provides an integer such that if and only if . Applications are given to lattice-ordered abelian groups with strong unit and to AF -algebras.

Abstract: Let G be an almost simple group with socle An, the alternating group of degree n. We prove that there is a unit of order pq in the integral group ring of G if and only if there is an element of that order in G provided p and q are primes greater than n 3. We combine this with some explicit computations to verify the prime graph question for all almost simple groups with socle An if n ≤ 17.

Abstract: Let R be a commutative noetherian ring. Denote by D−(R) the derived category of cochain complexes X of finitely generated R-modules with Hi(X) = 0 for i ≫ 0. Then D−(R) has the structure of a tensor triangulated category with tensor product ⋅⊗RL⋅ and unit object R. In this paper, we study thick tensor ideals of D−(R), i.e., thick subcategories closed under the tensor action by each object in D−(R), and investigate the Balmer spectrum Spc D − ( R ) of D−(R), i.e., the set of prime thick tensor ideals of D−(R). First, we give a complete classification of the thick tensor ideals of D−(R) generated by bounded complexes, establishing a generalized version of the Hopkins–Neeman smash nilpotence theorem. Then, we define a pair of maps between the Balmer spectrum SpcD−(R) and the Zariski spectrum SpecR, and study their topological properties. After that, we compare several classes of thick tensor ideals of D−(R), relating them to specialization-closed subsets of SpecR and Thomason subsets of SpcD−(R), and construct a counterexample to a conjecture of Balmer. Finally, we explore thick tensor ideals of D−(R) in the case where R is a discrete valuation ring.

Abstract: Let U be a strong monoidal functor between monoidal categories. If it has both a left adjoint L and a right adjoint R, we show that the pair (R,L) is a linearly distributive functor and (U,U)⊣(R,L) is a linearly distributive adjunction, if and only if L⊣U is a Hopf adjunction and U⊣R is a coHopf adjunction. We give sufficient conditions for a strong monoidal U which is part of a (left) Hopf adjunction L⊣U, to have as right adjoint a twisted version of the left adjoint L. In particular, the resulting adjunction will be (left) coHopf. One step further, we prove that if L is precomonadic and $L\mathbb {1}$ is a Frobenius monoid (where $\mathbb {1}$ denotes the unit object of the monoidal category), then L⊣U⊣L is an ambidextrous adjunction, and L is a Frobenius monoidal functor. We transfer these results to Hopf monads: we show that under suitable exactness assumptions, a Hopf monad T on a monoidal category has a right adjoint which is also a Hopf comonad, if the object $T\mathbb {1}$ is dualizable as a free T-algebra. In particular, if $T\mathbb {1}$ is a Frobenius monoid in the monoidal category of T-algebras and T is of descent type, then T is a Frobenius monad and a Frobenius monoidal functor.

Abstract: Abstract W is the category of archimedean ℓ-groups with designated weak order unit. The full subcategory of W of objects for which the unit is strong unit is denoted by W∗; such ℓ-groups are called bounded. Thus arises a coreflection . For G ∈W, Y G is the Yosida space, and G ≤ pG is the much-studied projectable hull. Recently, in [1], for G ∈ W∗, Y pG is identified as the Stone space of a certain boolean algebra of subsets of the minimal prime spectrum Min(G), and skepticism is expressed about extending this to W. Here, we show that indeed such an extension is possible, using a result from [5] and the following simple facts: in very concrete ways Min(G) and Min(BG) are homeomorphic spaces, and and are isomorphic boolean algebras; p and B commute.

Abstract: In categories of linear relations between finite dimensional vector spaces, composition is well-behaved only at pairs of relations satisfying transversality and monicity conditions. A construction of Wehrheim and Woodward makes it possible to impose these conditions while retaining the structure of a category. We analyze the resulting category in the case of all linear relations, as well as for (co)isotropic relations between symplectic vector spaces. In each case, the Wehrheim-Woodward category is a central extension of the original category of relations by the endomorphisms of the unit object, which is a free submonoid with two generators in the additive monoid of pairs of nonnegative integers.

Abstract: The concept of internally cancellable rings has been extensively studied in the literature. This paper seeks to continue the study of these rings and find some new characterizations. It is proved that R is “IC”, if and only if for each regular element a ∈ R, and idempotent element b ∈ R with Ra + Rb = R, there exists x ∈ R such that a + xb is a unit (alternatively, unit-regular element) in R and aR ∩ xR = 0. In case the ring R has the summand sum property, we indicate that R is IC, if and only if for each regular element a ∈ R, and element b ∈ R with Ra + Rb = R, there exists an idempotent e ∈ R, such that a + eb is a unit in R and aR ∩ eR = 0.

Abstract: Hans J. Zassenhaus conjectured that for any unit $u$ of finite order in the integral group ring of a finite group $G$ there exists a unit $a$ in the rational group algebra of $G$ such that $a^{-1}\cdot u \cdot a=\pm g$ for some $g\in G$. We disprove this conjecture by first proving general results that help identify counterexamples and then providing an infinite number of examples where these results apply. Our smallest example is a metabelian group of order $2^7 \cdot 3^2 \cdot 5 \cdot 7^2 \cdot 19^2$ whose integral group ring contains a unit of order $7 \cdot 19$ which, in the rational group algebra, is not conjugate to any element of the form $\pm g$.

Abstract: This X-ray CT device uses iterative reconstruction to obtain a CT image with the desired noise reduction or X-ray reduction ratio An iterative approximation reconstruction unit (136) is provided which, from measured projection data obtained by an X-ray detection unit of the X-ray CT device, iteratively reconstructs a CT image in a reconstruction range of a subject, and iteratively corrects the CT image such that calculated projection data, calculated through forward projection of a CT image, is the same as the difference between measured projection data detected by the X-raw detection unit and calculated projection data The iterative approximation reconstruction unit is provided with a parameter determination unit (151), an iterative correction unit (152), and a table unit (153) which calculates in advance the relation between each parameter used in the iterative reconstruction, and noise reduction or X-ray reduction ratio in the CT image The parameter determination unit (151) determines the parameters from a calculation table in the table unit (153) depending on the desired reduction ratio

Abstract: A ring R is called clean if every element of it is a sum of an idempotent and a unit. A ring R is neat if every proper homomorphic image of R is clean. When R is a field, then a complete ch...

Abstract: Anderson's theorem states that if the numerical range W(A) of an n-by-n matrix A is contained in the unit disk and intersects with the unit circle at more than n points, then it coincides with the (closed) unit dissk An analogue of this result for compact A in an infinite dimensional setting was established by Gau and Wu We consider here the case of A being the sum of a normal and compact operator

Abstract: Let $N$ be a nilpotent normal subgroup of the finite group $G$. Assume that $u$ is a unit of finite order in the integral group ring $\mathbb{Z} G$ of $G$ which maps to the identity under the linear extension of the natural homomorphism $G \rightarrow G/N$. We show how a result of Cliff and Weiss can be used to derive linear inequalities on the partial augmentations of $u$ and apply this to the study of the Zassenhaus Conjecture. This conjecture states that any unit of finite order in $\mathbb{Z} G$ is conjugate in the rational group algebra of $G$ to an element in $\pm G$.

Abstract: The lattice A1(U ∨(X)) of subalgebras with unit of the semifield U ∨(X) of continuous positive functions defined on a topological space X is considered. A topological space is said to be a Hewitt space if it is homeomorphic to a closed subspace of a Tychonoff power of the real line R. The main achievement of the paper is the proof of the fact that any Hewitt space X is determined by the lattice A1(U ∨(X)).

Abstract: Zassenhaus conjectured that any unit of finite order in the integral group ring $\mathbb{Z}G$ of a finite group $G$ is conjugate in the rational group algebra of $G$ to an element in $\pm G$. We review the known weaker versions of this conjecture and introduce a new condition, on the partial augmentations of the powers of a unit of finite order in $\mathbb{Z}G$, which is weaker than the Zassenhaus Conjecture but stronger than its other weaker versions. We prove that this condition is satisfied for units mapping to the identity modulo a nilpotent normal subgroup of $G$. Moreover, we show that if the condition holds then the HeLP Method adopts a more friendly form and use this to prove the Zassenhaus Conjecture for a special class of groups.

Abstract: Using the category of finite sets and injections, we construct a new model for the multilinearization of multifunctors between spaces that appears in the derivatives of Goodwillie calculus. We show that this model yields a lax monoidal functor from the category of symmetric functor sequences to the category of symmetric sequences of spaces after evaluating at the unit.

Abstract: This essay reports on a writing-based formative assessment of a university-wide initiative to enhance students’ global learning. Our mixed (and unanticipated) results show the need for enhanced expertise in writing assessment as well as for sustained partnerships among diverse institutional stakeholders so that public programming—from events linked to classroom-level learning to broader cross unit mandates like accreditation—can yield more rigorous, responsive, and mixed method assessments.